Average Error: 29.8 → 0.5
Time: 20.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}
double f(double x) {
        double r804160 = x;
        double r804161 = 1.0;
        double r804162 = r804160 + r804161;
        double r804163 = cbrt(r804162);
        double r804164 = cbrt(r804160);
        double r804165 = r804163 - r804164;
        return r804165;
}
↓
double f(double x) {
        double r804166 = 1.0;
        double r804167 = x;
        double r804168 = cbrt(r804167);
        double r804169 = r804167 + r804166;
        double r804170 = cbrt(r804169);
        double r804171 = r804168 + r804170;
        double r804172 = r804170 * r804170;
        double r804173 = fma(r804171, r804168, r804172);
        double r804174 = r804166 / r804173;
        return r804174;
}

Error

Bits error versus x

Derivation

  1. Initial program 29.8

    \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
  2. Using strategy rm

  3. Applied flip3--29.7

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}}\]

  4. Simplified29.1

    \[\leadsto \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

  5. Simplified29.1

    \[\leadsto \frac{\left(x + 1\right) - x}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}}\]

  6. Taylor expanded around 0 0.5

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}\]

  7. Final simplification0.5

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))